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Properties

Label 1-2015-2015.19-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.800 + 0.599i$
Analytic cond. $216.541$
Root an. cond. $216.541$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.309 + 0.951i)3-s + (0.104 + 0.994i)4-s + (−0.866 + 0.5i)6-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.994 − 0.104i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (−0.207 − 0.978i)18-s + (−0.743 − 0.669i)19-s + (0.743 + 0.669i)21-s + (0.809 + 0.587i)22-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.309 + 0.951i)3-s + (0.104 + 0.994i)4-s + (−0.866 + 0.5i)6-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.994 − 0.104i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (−0.207 − 0.978i)18-s + (−0.743 − 0.669i)19-s + (0.743 + 0.669i)21-s + (0.809 + 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.800 + 0.599i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.800 + 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.400879039 + 0.7995887178i\)
\(L(\frac12)\) \(\approx\) \(2.400879039 + 0.7995887178i\)
\(L(1)\) \(\approx\) \(1.189540406 + 0.7588065787i\)
\(L(1)\) \(\approx\) \(1.189540406 + 0.7588065787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
11 \( 1 + (0.994 - 0.104i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.207 + 0.978i)T \)
43 \( 1 + (0.669 - 0.743i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.207 - 0.978i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (0.406 - 0.913i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (0.743 + 0.669i)T \)
89 \( 1 + (-0.406 + 0.913i)T \)
97 \( 1 + (0.994 - 0.104i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.66866904227315995132004989969, −18.95284383788196589634153495968, −18.427928948620431990673452438361, −17.794813570878635573503621360752, −16.82242362209456856710784082087, −15.97842500466107927097771982744, −14.95113588319804610168112169091, −14.36166919773055359308357416207, −13.83133721948933565325206823745, −12.828865566430963825478827091774, −12.23162705328100841313909803207, −11.8314605923376602297886893071, −11.0995942567730152970324660938, −10.28248119450342217764655161915, −9.143071119774798118577570124824, −8.5847912830771818686897311765, −7.45696517857947658238176971478, −6.42557932175356208655882599391, −6.12652760127517893686746880861, −5.06427356688731739858432744598, −4.47547259168791661351298692041, −3.19054916320613607063147389936, −2.4186392728785158124199775489, −1.66605677527424530510942312383, −0.86137672805695555701683472069, 0.40045939866180330884958127817, 1.847841183485380495918627243138, 3.20633372960531322711394224192, 3.95803695310001241959135833130, 4.3785976295545466384850250196, 5.22080613239912528404618876749, 6.17701550898785765089412885215, 6.68778615156662299714092088889, 7.73574608970834244013426203190, 8.53198658068256050799371034503, 9.28725463359412447214018818146, 10.26595571035623788654729279088, 11.11954268802156616417293543883, 11.599564361009930523499086883010, 12.51610500031622380692709593699, 13.4706511300397893752883935339, 14.09298146979855973804594791492, 14.88184323989950789867964600607, 15.25284128324942456855664370481, 16.265599333921078563463789199575, 16.75532010406040474756516138826, 17.58356475730153199725460365557, 17.645159585057356841182396408684, 19.505448915241001619342045867220, 19.85354993369054162123734016665

Graph of the $Z$-function along the critical line